Treadle Sewing Machine Cabinet

I have a treadle sewing machine in the original cabinet (ca 1910's) that was damaged during shipment. The veneer on the lid was broken in several places and there was some water damage when the box got wet. A refinisher has quoted me $500 to refinish the entire cabinet (since the spaces with missing veneer are too large to match to the original, the veneer will be replaced; since the new veneer will not match the rest of the cabinet, the entire cabinet will be refinished). My thought before the damage occurred was to restore the finish, or perhaps strip the finish and reapply. Cleaning the surface with Murphy's Oil Soap leaves it a bit tacky, but doesn't seem to be having much impact on the appearance (the last chair I restored, the oil soap disolved the finish into black sludge - but not so this time around). I've tested the finish with nail polish remover, and it has no effect. I've read that in the absence of denatured alcohol, rum or vodka can be used to test the finish; rum (40%)had no effect. But isn't this piece too old to have varnish? Or can the original finish have completely worn off from use?

The cabinet is quite dark, and the surface is cracked, primarily where there was slight water damage. The rest of the cabinet has a somewhat uneven appearance colorwise.

How do I determine the type of finish? What should I use to clean the surface so I can determine if the finish must be removed or not? How do I proceed from there?

Thanks.